= Maximum growth rate of ideal magnetorotational instability
{title2=$s_{\max}=q\Omega/2,\quad\omega_{A,\max}^2=q(4-q)\Omega^2/4$}
For local cylindrical shear $q=-d\ln\Omega/d\ln r$ with $0<q<2$, maximizing the positive squared-growth root of the <ideal magnetorotational dispersion relation> gives the displayed formulas. For a Keplerian disk these are $s_{\max}=3\Omega/4$ and $\omega_A^2=15\Omega^2/16$. Field strength selects the fastest wavelength, while the maximum ideal rate is set by shear. Finite domains and magnetic diffusion can exclude that wavelength.
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